Rotational Motion About A Fixed Axis-Kinematics And Dynamics Notes:

1. Rotational Kinematics:

• Angular displacement (θ): The angular displacement of a rotating object is the angle through which it rotates about a fixed axis.
• Angular velocity (ω): The angular velocity of a rotating object is the rate at which it rotates about a fixed axis.
• Angular acceleration (α): The angular acceleration of a rotating object is the rate at which its angular velocity changes.

Equations of rotational motion (with constant angular acceleration):

• θ = ω₀t + ½ αt²
• ω = ω₀ + αt
• α = (ωf - ω₀)/t

Relationship between linear and angular motion:

• v = rω
• a = rα

Rolling motion:

• Rolling motion is a combination of rotational and translational motion.
• The point of contact between a rolling object and the surface it is rolling on is called the point of contact.
• The velocity of the point of contact is zero.
• The angular velocity of a rolling object is equal to the linear velocity of the point of contact divided by the radius of the object.

Reference: NCERT Class 11 Physics, Chapter 10: Rotational Motion

2. Rotational Dynamics:

• Moment of inertia (I): The moment of inertia of an object is a measure of its resistance to rotational motion.
• Torque (τ): The torque applied to an object is a force that causes the object to rotate about a fixed axis.

Rotational inertia and torque:

• The torque applied to an object is equal to the product of the moment of inertia and the angular acceleration:

$$τ = Iα$$

Parallel axis theorem:

• The moment of inertia of an object about an axis parallel to its center of mass is equal to the moment of inertia about the center of mass plus the product of the mass and the square of the distance between the two axes:

$$I = I_{CM} + Md²$$

Perpendicular axis theorem:

• The moment of inertia of an object about an axis perpendicular to its center of mass is equal to the sum of the moments of inertia about two perpendicular axes passing through the center of mass:

$$I = I_x + I_y + I_z$$

Conservation of angular momentum:

• The total angular momentum of a closed system remains constant:

$$L = Iω$$

Reference: NCERT Class 11 Physics, Chapter 10: Rotational Motion

3. Energy in Rotational Motion:

• Rotational kinetic energy (K_rot): The rotational kinetic energy of an object is the energy due to its rotation about a fixed axis:

$$K_{rot} = ½ Iω²$$

• Work done by a torque: The work done by a torque is equal to the product of the torque and the angular displacement:

$$W = τθ$$

• Conservation of energy: The total mechanical energy of a closed system remains constant:

$$K_{rot} + U = E$$

Reference: NCERT Class 11 Physics, Chapter 10: Rotational Motion

4. Gyroscopic Effects:

• Gyroscopic precession: Gyroscopic precession is the phenomenon in which the axis of rotation of a spinning object precesses (wobbles) about an external axis.

• Applications of gyroscopes: Gyroscopes are used in various devices such as compasses, navigation systems, and stabilizers.

Reference: NCERT Class 11 Physics, Chapter 10: Rotational Motion

5. Simple Harmonic Motion:

• Simple harmonic motion (SHM): SHM is a periodic motion in which the restoring force is directly proportional to the displacement from the mean position.

• Equations of SHM:

$$x = Acos(ωt + ϕ)$$

$$v = -Aωsin(ωt + ϕ)$$

$$a = -Aω²cos(ωt + ϕ)$$

• Energy in SHM: The total energy of a particle in SHM is constant and is given by:

$$E = ½kA²$$

• Relation between SHM and uniform circular motion: SHM is a projection of uniform circular motion on a diameter.

Reference: NCERT Class 11 Physics, Chapter 15: Oscillations

6. Damped and Forced Oscillations:

• Damped oscillations: Damped oscillations are oscillations in which the amplitude decreases with time due to friction or other resistive forces.

• Forced oscillations: Forced oscillations are oscillations that are caused by an external force that varies with time.

• Resonance: Resonance occurs when the frequency of the external force matches the natural frequency of the oscillator, causing the amplitude to become very large.

Reference: NCERT Class 11 Physics, Chapter 15: Oscillations

7. Motion in a Vertical Circle:

• Vertical circular motion: Vertical circular motion is a motion in which an object moves in a circular path in the vertical plane.

• Energy conservation in vertical circular motion: The total energy of an object in vertical circular motion is conserved:

$$K + U = E$$

Reference: NCERT Class 11 Physics, Chapter 10: Rotational Motion